Indexing Table with Backlash.

Mechanical Layout of the Indexing Table

Mechanical Layout of the Indexing Table

Table

20 workstations are equally spaced on a Ø500mm P. C. D. [Pitch Circle Diameter]
Each station has a mass of 1.5kg.
The table is steel and a diameter of Ø600mm and is 16mm thick.
The table is indexed by an 8-Stop Globoidal cam indexer through gears with a 2.5 : 1 reduction ratio.

The table is directly attached to the main shaft

It is Ø50mm x 320mm long.
The length of the main shaft that is in torsion form the underside of the table to the centre of the large gear wheel is 275mm.

Gears

The gears are Ø128mm and Ø320mm x 40mm wide, made of steel.

Indexer

The indexer turret is Ø163mm x 80mm thick. It has 8 rollers Ø2" x 1.25" long, on a pitched circle of Ø195mm.
The indexer turret is mounted on a shaft Ø50mm x 208mm long.
The length of the turret shaft that is in torsion is 137mm.

The cam motion-law is Mod-Sine, 180º index period, and runs at 150RPM.  The cycle time is 0.4s with 0.2s motion and 0.2s dwell.

We can assume the cam-shaft runs at constant angular velocity.

There is an estimated overall Friction Torque of 2.0Nm in the output transmission referred to the indexer turret.

The maximum backlash permitted in the cam track, when manufactured is 0.04mm.

The backlash in the gears can wear to 0.1mm before adjustment or replacement.


Mass and Inertia of Components

 

Mass (kg)

Moment of Inertia (kg.m2)

Slow Speed Assembly Tools

Work Stations - 1.5 × 20

30.00

1.899

Table : Ø600 × 16mm thick

35.28

1.588

Flange: Ø180 × 25mm thick

4.96

0.020

Main Shaft: Ø50 × 320mm long

4.90

0.002

Large Gear: Ø320 × 40mm thick

25.25

0.325

Slow Speed Total :

    100.41

3.834

High Speed Assembly

Small Gears: Ø128 × 40mm thick

4.02

0.008

Turret shaft Ø50 × 208 mm long

3.19

0.001

Turret body: Ø163mm × 80 mm thick

13.02

0.043

Roller Followers: Total of 8

4.02

0.039

High Speed Total :

24.25

0.091

Combined Total (referred to Indexer Turret) :

124.66

0.704

The mass and inertia of each component are calculated as shown in the table above.

The final inertia (Combined Total) total in the above table takes into account the gear ratio.

The inertia of the slow-speed assembly is reduced by the square of the gear ratio when referred to the indexer turret shaft.

Total Inerita Example ...an illustration of the benefits of a speed reduction!


Assessment of Dynamic System.

To assess the dynamic response, we estimate the natural vibration frequency of the system, and from that, the Period-Ratio.

The gears have a significant mass and inertia in an intermediate position in the transmission, which means that the system will not vibrate with a single, simple frequency, as required by the foregoing dynamic response theory!

However, the high inertia of the table and work stations imposes a dominant frequency which gives a good approximation to the theoretical model: this is nearly always the case in practice. But, be aware.


Rigidity

Torsional Rigidity of Shaft : Shaft-Stiffness ;

Bending Stiffness between bearings : TorqueGear16

Rigidity: R = S × r2


Turret Shaft Torsional Rigidity

Eqtn-r1

Second moment of area of shaft is:

eqtn-j1

Thus: Main Shaft Bending Stiffness is:

eqtn-s1

Note: Radius of Large Gear = 0.16m:

Its equivalent Torsional Rigidity is:

eqtn-r2


Indexer High Speed Shaft Torsional Rigidity

eqtn-r3

Turret Bending Stiffness is:

eqtn-s2

Note: Radius of Small Gear = 0.064m

Its equivalent Torsional Rigidity is:

eqtn-r4


The combined rigidity, referred to the indexer turret:

eqtn-rt

[Note that the bending stiffness is so high it could have been ignored].


Approximate Natural Frequency

eqtn-nf1

eqtn-nf2

Natural Period = 1/f = 0.0321 seconds [assuming a single degree of freedom].


Period-Ratio

The motion period is 0.2 seconds.

The Period-Ratio is:

n = Motion Period / Natural Period = T / [1/f] = f × T

n = 31.17 × 0.2 = 6.234


Nominal Peak Inertia Torque

The total inertia referred to the indexer is

InertiaReferred

Peak Angular Acceleration:

The output stroke of the indexer is:

StrokeRadians

Index Period of the Indexer is:

IndexPeriod

The Coefficient of Acceleration of the Mod-Sine is:

ModSineCa

Nominal Peak Acceleration is:

NominalAcc

AccRationNom

Nom-PeakInertiaTorque

Peak Inertia Torque after Torque-Factor

Torque-Factor

We can use this equation to find the Torsion-Factor

Torsion-Factor Equation;

The Parameters for the Mod-Sine Motion-Law are:

Torsion-Factor-ParameterP ; Torsion-Factor-ParameterQ ; Torsion-Factor-ParameterR

The Period-Ratio, n. is 6.34

Using the parameters, the Torsion-Factor is

Torsion-Factor

We must increase the Nominal Peak Inertia Torque by the Torsion-Factor:

PeakInertiaTorque

Add Friction Torque

We must add the Friction Torque for the mechanism, ignoring the backlash impact effect.

PeakInertiaTorque-Friction


Include Backlash

We must consider the impact shock load after the transition of backlash...

Total Backlash, expressed as an angle (radians), at the Indexer Turret.

Gear Teeth, 0.1mm, @ 64mm radius:

Backlash-gears

Indexer Turret, 0.04mm @ 97.4mm radius:

Backlash-IndexerTurret

Total Backlash

TotalBacklash-IndexerTurret

Normalized backlash [backlash against angular stroke]

NormalisedBacklash-IndexerTurret

Deceleration

The natural deceleration of the system due to the deceleration torque on the payload during 'Free-Flight' is:

natural-deceleration

Normalized Deceleration is:

NormalisedDecel

This is quite low and can be taken as Zero!

Normalized Impact Velocity

With a Normalized Backlash of 0.00251 and Normalized Deceleration of 0.0, then:

Normalized Impact Velocity:

NormalisedImpactVelocity

Real Impact Velocity:

RealImpactVelocity

Peak Shock Torque

The Peak Shock Torque on the Turret is:

PeackShockTorque

At 40% of the Peak Inertia Torque without Backlash, this is a significant load, and should not be ignored in the design of the mechanism.

A safe way of taking it into account is simply to add it to the peak vibration torque (this assumes the two peak torques occur at exactly the same point in the motion, which is quite possible):

Peak Torque at Output Shaft.

Add the Peak Shock Torque to the Vibration Torque

PeackTotalTorque

Although it could be argued that this is too pessimistic, it does illustrate that to design the mechanism on the basis of the Nominal Dynamic Torque of 78.41Nm would be under-estimated!

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